Tag: Technology

In the recent posts we have been focusing on the statistics elements of the new course. It is now time for an update on the general running of the course and the pure elements.

Firstly some background information on the new A-level group. In previous years we have offered the maths A-level in two different option blocks meaning that we have had two groups of around 10 – 12 students. With drop off as students have dropped from four to three subjects at the end of year 12 this has meant groups of 5 – 8 in year 13. This has now been considered uneconomical so we have been reduced to one option block, as it was thought impossible to combine the two groups at the end of year 12. Of course having had this decision imposed on us, we then had a greater take up of maths meaning a class of 26. The balance of this class is also unusual due to a foreign exchange programme meaning that we have a large number of foreign students joining us for the year (two Swedes, two Mexicans, one American and one Romanian). These students will leave at the end of year 12 leaving a much more manageable year 13 group.

In practical terms the sharing of the content evenly has gone extremely well, with us being able to have frequent conversations about the direction of the lessons, despite the added pressure on my time due to being head of department this year. This has been even better than expected, as having the dialogue has meant (certainly for me and I hope for Will), that the actual planning of the lesson has been easier as I already know what I want to achieve. We have been careful to make sure that both of us have taught both pure content and statistics content to avoid being pigeon-holed as the ‘pure’ teacher and the ‘stats’ teacher.

Another positive has been the use of technology in lessons. In the first three weeks of term I have already booked and used a computer room more than I had in the previous two years. All of the students have now got their new calculators and we are settling into using them – I even treated myself to a new one, replacing the one I had used since I did my A-levels 16 years ago.

Our first lesson looked at extending GCSE proof. We felt that it was really important to introduce proof as one of our key themes as early as possible. We asked students to choose from a variety of ideas, both algebraic and geometrical, to see what they could come up with. We then discussed what a proof should look like and worked on developing the skills required to build a mathematical argument. This is something that we will be returning to regularly, making sure that students are getting more accomplished.

Our first major topic was indices and surds. As this is largely revision of GCSE topics we decided to approach it by setting pre-learning tasks and then developing the knowledge already in place. The tasks we set were the ‘walkthroughs’ from Integral Maths. These walk students through the required knowledge of the topic, introducing ideas and allowing experimentation in an interactive way. Students are allowed as many attempts as required without it being recorded and reported to teachers.

For the indices section of the week we used the online textbook ‘Problem Book for A-level Maths’ being developed by Stuart Price (@sxpmaths on Twitter). We really like the way that this has sections for students who are at different stages of development, starting with technique for those who need more work on the basics and progressing through problem solving to puzzles & challenge. Our grade 9 students had great fun going straight onto the challenge problems, while others were absorbed by the technique section. They were also able to move backwards and forwards if required. There is also a final section for each topic which covers exam style question practice, although we did not get to this – something to use in the future.

For the surds section we used an adaptation of an activity from Integral Maths (pictured below) where students discussed how surds are simplified. We deliberately left some expressions that had not been fully simplified and asked students to present their findings to each other. Additional questions for this lesson were taken from Dr Frost Maths (@DrFrostMaths).

We have been really enjoying using materials from a variety of sources and with different styles. Students seem really enthused and have been seeking us out for extra support when needed. Our next topic is exponentials and logarithms. The logic behind this is that we wanted students to experience something totally new early in the course. We also felt that they fitted very well together, essentially being two different ways of looking at the same topic. I always felt that the two ends were rather artificially kept apart by the arbitrary barrier between C1 and C2, now I have the opportunity to try and teach them together.

Will Davies has been working with us on the scheme of work for the new A-level. Over the last few years he has predominantly taught the statistics content for the A-level courses. Here are his thoughts on the large data set:

“When the new specifications were announced the introduction of these “large data sets” (LDS) left me sceptical, and unsure of exactly how we were going to work with them. With time came a lot more clarity; actually being able to pick over the data sets that were released with the sample assessment material meant we could start to see how they were going to be assessed, and how they might fit into our teaching.

And I have come to this conclusion: the LDS is my joint-favourite thing about the new A-Level – the other aspect being that we’ve been able to tear up the old order of topics and build a curriculum that we feel teaches maths in the most logical order and in the best manner. Being able to combine the applied topics in with the pure topics they depend on is key: e.g. binomial theorem and binomial expansion, as well as teaching variable acceleration immediately after calculus is taught.

I have read on Twitter a lot of negativity about the LDS, and I am unsure why. My instinct says that the reason is because the LDS is being perceived as a separate topic that needs to be taught in addition to other content (that we’re already unsure whether we can fit it all in satisfactorily). As a department, from very early in the process we realised that this shouldn’t be the case – the LDS is not a separate topic, it is instead the tool that you use to teach all the data-handing parts of the course.

Every time you do an example – it comes from the LDS.

Every time you set an exercise in class – it comes from the LDS.

Every time you set a homework – it comes from the LDS.

The more the students immerse themselves in the LDS the more familiar they become with it. Homeworks can be to do some calculations or create some charts (and email them to use in advance where appropriate) then as a group we can discuss next lesson. My other big idea for embedding the LDS into our lessons is to have at least once per week a Show-me / Tell-me starter (regardless of whether the lesson is going to be on stats or not). Students will be encouraged to do a little investigation themselves, then getting the class to discuss together discuss the potential causes (e.g. our outliers). This will be way in which we can as a class build up a bank of interesting observations of our LDS, just like the observation we made when we were examining the MEI sample assessment material.

This question from the MEI sample A Level assessment – we were drawn to the very long tail at the bottom of the Sub-Saharan Africa box-plot, and wondered which countries were causing this. Looking at the LDS we quickly came up with 3 countries with very low birth rates: Saint Helena, Mauritius, Seychelles – all island nations. Which feels like a nice fact – that the island nations of Sub Saharan Africa have significantly lower birth rates than other countries in that region.

This brings me onto our choice of exam board – the data sets are not provided in the exam, yet students are expected to be able to use some very specific knowledge of them in order to gain some marks in their exams. With the large LDSs (like Edexcel’s weather data) you could study that for a couple of years and maybe still have examined at the key pieces of data.

So, MEI has the smallest large data set (covering information about the 237 countries of the world), and that brings its own advantages – it is printable. The bulk of it fits on 3 A3 pages, and I have created a single A4 page that expands on the Dependency status of relevant countries. So now all our students have a hard copy of their data set to use – meaning that we don’t always have to be in an IT room when we’re working on it. The other major advantage is that on presenting students with the data set they immediately felt that because it actually wasn’t “too big” that knowing it well was going to be achievable.

When it comes to using technology there are various ways in which we plan to incorporate this with the LDS. The ClassWiz calculator is clearly going to be key as, as is learning a bit about Excel. Filters, sorting and a deep look into the inbuilt statistical formulae will all need to take place – not just for the sake of the LDS, but Excel skills are incredibly useful. We’re also going to look to support/enhance teaching & learning by graphing some of the data in Geogebra and Gnumeric. (Gnumeric is apparently a very good tool for creating box-plots although I am yet to explore that any further). I have also built in Excel a sampler tool that will create random samples from the LDs, although it still needs perfecting. When it is complete I will share it here.

When it comes to assessments, starting work on the LDS from lesson 2 means we will be able to include it in assessments from half term 1 – to start with we will make sure we write the assessments so we know that students have seen (in one form or other) what we will be asking about, then we can progressively choose more and more obscure statistics to include. Finally we plan to set students extended projects to do. These like likely asked them to choose some aspect of the data set, be it a group of regions or a groups of fields, calculate some statistics, create some charts, draw some conclusions, and to write up a little report on their findings.

Identifying the smallest data set, and revisiting it weekly for 2 years will give students the best chance of becoming as familiar as they can be with the LDS (aside from dedicating too much curriculum time to it). I suppose the bottom line is that we feel that using the LDS to teach all data topics is going to be such an improvement on using (essentially random) examples that are using a similar approach with our GCSE statistics. In lessons our year 9s and 10s are currently populating their own data-set (containing information about themselves). They have really enjoyed the data collection (although I did receive a complaint from the English classroom underneath the standing long-jump) – now to analyse it!”

In our work on revamping the curriculum for the new specification we have been careful to make sure that we are considering the overarching themes of problem solving and use of technology. We are very keen to ensure that use of technology does not become ever more complicated and ‘interactive’ PowerPoint files that are demonstrated from the front of the class room with little chance for students to use and develop their own skills. We also want to introduce this aspect as early as possible to encourage students to think about technology as a vehicle for working on and solving problems when they get stuck. On Friday I met with Simon, Fiona Kitchen (from FMSP) and two colleagues from my department to discuss methods for this.

Our starting point was to use the worksheet “Problem Solving with Geogebra” from MEI’s scheme of work. We looked at solving the problems ourselves, trying to limit techniques to those that year 12 students were able to use. This proved rather difficult! After much wrestling (and a plea to twitter) we managed to create working models in Geogebra for the first three of the problems.

We had a lot of fun working on these problems but concluded that the level was a too high for students who are starting out in year 12. As such we will need to adapt to something closer to GCSE if we are going to introduce Geogebra in this way at the beginning of the course. One thing that struck me during the afternoon was that we persevered with the problems for a long period, around 2.5 hours. This was something that our students would have really struggled to achieve. The same morning one of my year 11 students, when confronted with a difficult question, said “It’s alright for you sir, you are good at maths and can do it easily.” I was unable to make her understand that I don’t find all maths easy and that I enjoy the struggle with harder problems. This perhaps sums up the major problem we have been fighting against with our A-level students over the last few years – the lack of resilience as soon as a problem gets complicated.

Hopefully this process of really concentrating on both problem solving and technology will help to improve this, which is certainly the focus of what we are looking at during this process. For now though here are the problems that we worked on.

Problem 1

With this problem I found it very easy to create a polynomial fixed by the points A, B and C. This initially created a point D that moved as A, B and C moved. To do this I used the measuring tool in Geogebra to calculate how far away the points were from the origin.

My second attempt used a division at the start of the polynomial to allow me to control point D as well. It was at this point that I realised the shortcomings of my method of measuring distances – when I moved the points to negative values Geogebra continued to measure them as positive. To complete the problem Simon showed me how to use just the x-ordinate of A etc. in the calculation. My final solution is at: https://www.geogebra.org/m/bZD6fARj

Problem 2

For this problem I started by drawing a circle centred on the origin with a point on the circumference fixed into the side AC. I then created another circle centred on C which connected to the first. I repeated this methodology to create a third circle centred on point B. The three circles could then be manipulated together until I had a solution that worked. This did not satisfy me – I wanted to be able to change the triangle and the circles to remain a solution to the puzzle.

My instincts for this puzzle were probably from spending time playing the mobile phone game Euclidea – maybe those hours were not completely wasted! I guessed that the points where the circles met on the edges of the triangle were those where the circle inscribed in the triangle also touched the sides of the triangle. My resulting solution can be found here: https://www.geogebra.org/m/jMSThZg8

Problem 3

Problem 3 caused me the most problems. For a long time I was able to either create a line that was perpendicular to the tangent from A or a line that passed through the point B but not both. After a long time trying (and a plea to Twitter), Simon came up with a solution that can be found at: https://www.geogebra.org/m/MFgcJaAd

Problem 4

We ran out of time before tackling problem 4 and I have not yet returned to it. I have some ideas about using a quadratic function with roots that are the x-ordinates of A and B and integrating it but have not progressed any further yet. I leave that one with you…